Nonparametric Estimation of a Distribution Function in Biased Sampling Models

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Abstract

The nonparametric maximum likelihood estimator (NPMLE) of a distribution function F in biased sampling models have been studied by Cox (1969), Vardi (1982, 1985), and Gill, Vardi, and Wellner (1988). Their approaches are based on the assumption that the observations are drawn from biased distributions of F and biasing functions do not depend on F. These assumptions have been used in Patil and Rao (1978). This thesis extends the biased sampling model by making the biasing functions depend on the distribution function F in a variety of ways. With this extension, many of the existing models, including the ranked-set sampling model and the nomination sampling model, become special cases of the biased sampling model. The statistical inference about F becomes to a large extent the study of the biasing function. We develop conditions under which the generalized model is identifiable. Under these conditions, an estimator of the underlying distribution F is proposed and its strong consistency and asymptotic normality are established. In certain situation, estimation of Fin a biased sampling model is in fact a problem of estimating a monotone decreasing density. Several density estimators are studied. They include the nonparametric maximum likelihood estimator, a kernel estimator, and a modified histogram type estimator. The strong consistency, the asymptotic normality, and the bounds on average error for the estimators are studied in detail. In summary, this thesis is a generalizations of the estimation results available for the ordinary s-biased sampling model, the ranked-set sampling model, the nomination sampling model, and a monotone decreasing density.